The present work gives explicit analysis for determining the ordinary and the dark energy density of the universe [1] [2] based on an unorthodox combination of modern extension of hyperbolic geometry [3] [4] and recent advances in nonlinear dynamics, deterministic chaos and random fractals [5] . We start from two parallel lines of thinking, namely first from our topological conception of a pre-quantum particle described by a bi dimension

zero set [6] [7] and a pre-quantum wave modelled by empty set [6] [7] where as found from the generic von Neumann-Connes dimensional function of an x manifold corresponding to Penrose fractal tiling universe [6]

The second line of thinking within the present scheme is the compact 4D hyperbolic manifold M⁽⁴⁾ of M.W.

Davis [4] which has 1400 cells and a hyperbolic volume equal [4] . Incidentally the Euler characteristic of this manifold is which differs from that of E-infinity Cantorian spacetime [6] - [14] by only being where k is the ‘tHooft renormalon [11] [15] . The possibility of a Hardy entangleon elementary particle as well as an entangleon particle was proposed by the

present Author as well as others a relatively short time ago [11] [15] - [17] . From here onwards and as we will see momentarily, the two lines of though mentioned will converge towards a common conclusion, namely that the holographic boundary of our universe can be modelled exactly by degrees of freedom of a compactified Klein modular curve plus gravity and spacetime [6] - [11] and that the exact ordinary energy density of the cosmos is given by [8]

where R⁽⁴⁾ = 20 is the number of independent components of the Riemann-Einstein tensor in D⁽⁴⁾ = 4 dimensional Einstein spacetime. Inserting in E(O) of equation (2) one finds the obvious result [8]

That means E(O) is about 4.5% of the maximal energy density of the cosmos, i.e. Einstein’s density while the dark energy density is clearly the self explanatory formula which is nearly 95.5% of Einstein’s maximal energy density E = mc² as found in earlier publications and in full agreement with the actual cosmic measurements [1] [2] [7] . Our next task is to give the details of the analysis leading to the preceding result using the suggested methodology of E-infinity theory [6] - [9] [11] - [15] [18] [19] .

M.W. Davis found in [4] a compact hyperbolic 4-manifold built upon 120-cell PCH⁽⁴⁾ where P can be cut into 14,400 congruent orthoschemes, each with a volume equal to. Consequently we have the total hyperbolic volume [3] [4]

Now the transfinite harmonization methods of E-infinity theory [6] - [9] suggest immediately upon inspecting

the above expression that 104 should be replaced by 104.7213596, by 10 and 3 by where is ‘tHooft’s renormalon [11] [15] which is twice the value of Hardy’s entangleon

[12] - [14] [20] . Now for the reader not familiar with E-infinity transfinite correction, the simplest thing to make the above plausible is to compute the expression of equation 4 first explicitly [3] [4]

Subsequently alone from the magnitude of the integer value 342 we notice that (342)(2) = 684 which is almost the value of the sum of all the dimensions of the two and three Stein spaces, namely [9] [18]

This result in turn is shown in E-infinity theory to be related to the theoretical exact value of the inverse electromagnetic fine structure constant by [12] - [14]

where and is Hardy’s quantum entangleon [11] - [17] [20] [21] . Thus the exact transfinite value for M⁽⁴⁾ could be surmised from the above to be

as we concluded earlier on. Confining ourselves to the integer theory only, it is easily reasoned that the 342.7 is approximately equal to 343 the hyperbolic volume of our manifold could be seen as the dimensions of a Klein modular space with its well known 336 degrees of freedom when we add to it the 7 embedding dimension of and it becomes 336 + 7 = 343 [7] [9] . Now we can see another novel interpretation of Vol_H(exact) of M⁽⁴⁾, namely by extracting the embedding it becomes [11] [15] :

which is simply equal to the compactified fractal degrees of freedom of with a transfinite number of isometries equal to [12] - [14]

where 336 may be seen as say the dim SL(2,7) or R⁽⁸⁾ of Riemannian tensor in super space D = 8 or alternatively the number of kissing spheres in ten dimensions. In our case we take 338.8854382 as being the

compactified holographic boundary of our actual spacetime which in an integer approximation of the standard model amounts to 336 plus |SU(2)| = 3 giving us the well known 339. Recalling that the first massless level in Heterotic string theory [16] is given by 8064 and that this may be found either from the bulk or alternatively from the holographic boundary by a multiplication of the degree of freedom of and the instanton number n = 24 [8] [9] [12] - [14] [16] leading to N_o = (336)(24) = 8064 then knowing that the exact transfinite number corresponding to 8064 is, in E-infinity, equal to N_o = (336 + 16k)(24 + 2 + k) = 8872.135957 [9] . From this relation and remembering that c = n for the fuzzy Kähler modelling E-infinity spacetime and taking all the previous results on board, it follows then that the transfinitely corrected M⁽⁴⁾ will also have the Euler characteristic c = 26 + k rather than c = 26 [3] [4] . In other words the transfinite version of M⁽⁴⁾ is a fuzzy manifold in the sense of E-infinity theory and consequently E-infinity fuzziness makes the theory more accurate through subjecting our manifold to the same rules of homology and co-homology which we used for our E-infinity Kähler manifold.

Consequently by squaring one finds

From the above we see that we are back to the same holographic boundary of our theory and that represents an important Betti number of our manifold of cohomology [6] - [8] [16] while is basically

the Immirzi parameter of our analysis which may be interpreted as the entanglement probability of two self entangled points in our space, each with self entanglement probability. This is the inverse of the expectation value of the Hausdorff dimensions of E-infinity spacetime, i.e. [9]

This way we see that our analysis is circulatory consistent.

Now we are in a position to tackle the task of determining the energy density of ordinary energy and consequently that of dark energy of the cosmos. Using the so far obtained insight into the “fuzzy” or transfinite M⁴ it is not difficult to reason that the maximal E of Einstein density was based on 4D space. That would mean that

degrees of freedom of the theory were not considered in relation to 336 + 16k minus 16

of the holographic boundary of our theory. Since E = mc² is the maximal one hundred percent energy density, then writing means that E_max is given by. We may stress at this point that looking at E = mc² as the maximal energy density possible in the cosmos is a crucial point in our theory and deceptively simple and is far from being trivial. On the other hand assuming that E-infinity spacetime and consequently M⁴ is our real spacetime, then should not be but [8]

Consequently we have to expect the measured ordinary energy density to be [11] - [15]

This then implies a “missing” dark energy density of

Setting k @ 0, this simplifies to our previously obtained results [11] - [15]

Note here, as elsewhere, how k plays a crucial role in smoothing the analyses and easing the reaching of general conclusions and deeper insight. In a manner of speech we could say that we put the entire problem under a transfinite microscope enabling us to see how ‘tHooft’s renormalon hypothetical elementary particle works [11] [15] . This normalon is actually made of two other hypothetical particles, namely Hardy’s entangleon [7] [8] since

Pondering this situation we see that the real power of transfinite correction is the following: while overwhelmingly integer based theories are to a far extent consistent, we notice that we have many different theories leading to similar results although they differ substantially from each other at various stages. Transfinite correction fuses different theories and shows that they are exactly the same and that everything fits with everything else at all stages of the analysis. We will address the same subject at various occasions in the rest of the present paper.

Seen in the conventional way the standard model contains twelve messenger particles given by the symmetry breaking ofSU(3) SU(2) U(1) dimensions. The experimental discovery of the Higgs is not included in the above, nor is the not discovered yet graviton let alone all super symmetric partners. However seen under the transfinite microscope, E-infinity theory tells us something quite if not radically different. The reason for this deviation is that E-infinity extends fuzziness, i.e. transfiniteness to the counting of particles. It is ordinary counting but with a fractal-Cantorian weight linked to it. This so called fractal logic of counting particles mutates the standard model to a truly magical structure by the following transformation from integer to transfinite counting were we will write on the left hand side the classical counting vis-à-vis the transfinite weight counting on the right hand side of the following Table 1 [17] .